Injective Hulls
The injective hull of a graph G is a unique smallest Helly graph H(G) into which G isometrically embeds. Here are a few graphs embedded into their precomputed injective hulls. Some graph parameters are calculated, such as the Helly-gap of each graph and the number of vertices and edges in G and H(G).
Cycles and their relatives
- G = C_4
- G = C_5
- G = C_6, and its complement
- G = C_7
- G = C_8, and powers G^2, and G^3
- G = C_9, and powers G^2, G^3, and G^4
- G = C_10, and powers G^2, G^3, and G^4
- G = C_11, and powers G^2, G^3, G^4, and G^5
- G = C_12, and powers G^2, G^3, G^4, and G^5
A few known graphs
- 3-fan
- Hypercube Q_3
- Krackhardt-Kite and its power G^2
- Petersen
- Chvatal
- Icosahedral
- G = 2C_4 + 2C_3, and its power G^2
- G = 2C_5 + 2C_3, and its power G^2
Other exemplary graphs
- A permutation graph
- A Helly graph G whose minor G’ has a larger Helly-gap
- G = C_4 + C_6, and powers G^2 and G^3
- 3-fan with K_4
- Opposite vertices of induced C_6 share a neighbor
- C_6 with false twin
- Four extreme vertices of isometric C_8 are 2-dominated by a central vertex
Small graphs
These are not described as they are more easily viewed.